Speed is defined as “The time rate of change of position of a body without regard to direction; in other words, the magnitude of the velocity vector.” (*Dictionary of Physics*, 3rd edition, McGraw-Hill, 2002.

This is ambiguous, however. Consider a light beam reflected off a surface:

(1) Since the light returns to its starting point, the total travel distance is zero, so the overall velocity is zero and the speed is zero.

(2) However, the interest is in each leg of the journey. In that case, in the first leg light travels +*L* in time *t*, and in the second leg light travels –*L* in time *t*. The mean velocity in the first leg is *v*_{1} = +*L*/*t*, and the mean velocity in the second leg is *v*_{2} = –*L*/*t*. The mean velocity for both legs is the harmonic mean of these two velocities because what is fixed and independent is the length, not the duration.

1/((1/*v*_{1}) + (1/*v*_{2})) = 1/((1/*L*) – (1/*L*)) = 1/0 = ∞.

Thus the mean velocity is infinite, and the mean speed of light is infinite.

(3) Another approach looks at length of each leg apart from direction. In that case, in the first leg light travels *L* in time *t*, and in the second leg light travels *L* in time *t*. The speed in each leg is *L*/*t*, so the mean speed of light is *L*/*t*. This is the best known approach to the speed of light.

It’s interesting that (2) leads to the Galilean transformation, and (3) leads to the Lorentz transformation.